What is the integral of (cot(t)csc(t))/(csc^7(t)) ?

The integral of (cot(t)csc(t))/(csc^7(t)) is (sin^6(t))/6+C

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integral of (cot(t)csc(t))/(csc^7(t))

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

\frac{sin ^{6} ( t )}{6} + C

Solution steps

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

Rewrite using trig identities

= \int sin^{5} (t) cos (t) d t

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{u ^{6}}{6}

Substitute back

u = sin (t)

= \frac{sin ^{6} ( t )}{6}

Add a constant to the solution

= \frac{sin ^{6} ( t )}{6} + C

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What is the integral of (cot(t)csc(t))/(csc^7(t)) ?
The integral of (cot(t)csc(t))/(csc^7(t)) is (sin^6(t))/6+C